SERIES REPRESENTATIONS AND SIMULATION OF ISOTROPIC RANDOM FIELDS IN THE EUCLIDEAN SPACE

This document represents the Pytho codes associated with Series Representations and Simulations of Isotropic Random Fields in the Euclidean Space by Zhengwei Ma and Chunsheng Ma

Setup

Here we set some initial values for simulation, as well as create some helper functions

(i) Example 3.3 (continued), Case I: \nu=0

We want \begin{equation} y_{1} = -log(U_{0, 1}) \end{equation}

\begin{equation} y_{2} = U_{log(2), log(8)} \end{equation}\begin{equation} v_{1} = y_{1}^{0.5} * exp(\frac{-y_{2}}{2}) \end{equation}\begin{equation} w_{1} = U_{0, 1} \end{equation}\begin{equation} u_{1} = U_{0, 1} \end{equation}

(ii) Example 3.3 (continued), Case II: 0 < \nu < 1

We want \begin{equation} y_{1} = \Gamma(0.5, 1) \end{equation}

\begin{equation} y_{2} = U_{2^{0.5}, 8^{0.5}} \end{equation}\begin{equation} v_{1} = \frac{y_{1}^{0.5}}{y_{2}} \end{equation}\begin{equation} w_{1} = U_{0, 1} \end{equation}\begin{equation} u_{1} = U_{0, 1} \end{equation}

(iii) Example 3.3 (continued), Case III: \nu = 1 or \nu > 1

We want \begin{equation} y_{1} = \Gamma(3, 1) \end{equation}

\begin{equation} y_{2} = U_{\frac{1}{64}, \frac{1}{4}} \end{equation}\begin{equation} v_{1} = y_{1}^{0.5} * y_{2}^{0.25} \end{equation}\begin{equation} w_{1} = U_{0, 1} \end{equation}\begin{equation} u_{1} = U_{0, 1} \end{equation}

(iv) Example 3.5 (continued)